Donald Trump : They won't do that without our approval. They do nothing without our approval.

Lots of Korean Media translated it as "South Korean government are not allowed to do it without getting permission from the US".

but I think it's wrong translation and this related some cultural differences.
I understand like, US and SK keep in touch each other and so US know and strongly believe SK will step forward about this issue with the consent of all.
Am I wrong?

I'd like to hear how native speakers think.
the word choice "approval" is too much for other countries and insulting word for Koreans? like sub-country?

It's a poor translation. He's saying the SK gov't is always deferential to the US on such matters. It's an observation of past behavior, not a mandate, as the translation implies.
– jimm101Oct 11 '18 at 12:49

1

Unfortunately, it does sound to me as though Trump is saying that the South Korean government would not take concrete steps toward any important diplomatic initiative or agreement without the approval (best case: support; worst case: permission) of the Trump administration. The sentiment expressed seems pretty demeaning to me—just as would the reverse case, if, say, Vladimir Putin said of Donald Trump, "He won't do that without our approval. He does nothing without our approval."
– Sven YargsOct 12 '18 at 4:47